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This is not directly a consequence of Böhm's theorem. Böhm's theorem states that for any strongly normalizing $A$ and $B$ that are not $\beta\eta$-convertible, there exists $\Gamma$ such that $\Gamma A = K$ and $\Gamma B = S$. You don't get to constrain $\Gamma = A$ or the required form $\lambda x. x U V$. There may be a way to use Böhm's theorem on a different term, but this particular system of equations can easily be solved manually.

If such a term exists then $\Delta K = K U V = U$, so to satisfy the second equation it is necessary that $U = S$.